Toward a thermodynamically consistent picture of the phase-field model of vesicles: Curvature energy

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 78, Issue 3, 2008, Pages

  Jamet, D ^a   Misbah, C ^b  

^a CEA GRENOBLE   (France)

^b UNIV GRENOBLE ALPES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

BENDING FORCES; CONSTITUTIVE LAWS; PHASE FIELD MODELLING;

BENDING (DEFORMATION);

EID: 51349114290     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.78.031902     Document Type: Article

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21. If one has in mind a more general form of the Gaussian contribution f (H) (with f an arbitrary nonlinear function of H), then the topological invariance does not hold anymore. We shall not, however, consider this general case. Note also that even with the linear function H, the topological invariance is not fulfilled in the strict sense within phase field, since we would have H | φ | dV, and not just HdA, where dA is the area element. However, in the asymptotic limit where the interface width tends to zero, we should recover the topological invariance.
22. Actually, the mean curvature is defined by s n, where s is the gradient along the contour (s refers to surface). However, since |n| =1, it is straightforward to show that s n=?n.
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33. While a surface tension does exist for fluid interfaces, the authors in were interested in not including this phase-field surface tension in the phase-field evolution itself, but rather in the momentum balance equation.
34. It is straightforward to show that n (s n) =0 and (s n) n=0 so that the determinant of s n, which is the third invariant, is nil. If we set A s n, then the ith components in the products above are defined as (nA) i = nj Aji, (An) i = Aij nj.